On the Spectral Characterization of Prime Coprime Graphs Associated with Dihedral Groups
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Keywords:
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Prime Coprime Graph, Dihedral Group, Laplacian Matrix, Normalized Laplacian Matrix, Seidel Matrix
Abstract
Spectral graph theory and group theory help reveal how the structure of graphs shapes their mathematical properties. This research aims to analyze the spectral descriptors of the prime coprime graph of a dihedral group associated with the Laplacian, normalized Laplacian, and Seidel matrices. We also establish a method for determining the eigenvalues of these matrices. The results show that the Laplacian, signless Laplacian, and normalized Laplacian energies of the obtained graph are always even integers, whereas the adjacency and Seidel energies are never odd integers. These findings are consistent with previous studies and further strengthen the role of spectral descriptors in understanding algebraicgraph structures.
References
Abdurahim and M. U. Romdhini (2026). Investigating Some Properties of Prime Coprime Graph for Dihedral Groups. Journal of Applied Mathematics and Informatics. In review
Adhikari, A. and S. Banerjee (2022). Prime Coprime Graph of a Finite Group. Novi Sad Journal of Mathematics, 52
Al-Tarimshawy, A. S. A. and J. Siemons (2021). Singular Graphs with Dihedral Group Action. Discrete Mathematics, 344(1); 112119
Aschbacher, M. (2000). Finite Group Theory. Cambridge University Press
Bapat, R. B. and S. Pati (2004). Energy of a Graph Is Never an Odd Integer. Bulletin of Kerala Mathematics Association, 1; 129–132
Brouwer, A. E. andW. H. Haemers (2011). Spectra of Graphs. Springer
Codogni, G. and G. M. Lido (2026). Spectral Theory of Isogeny Graphs. Journal of Number Theory, 286; 131–184
Cole, K. D., A. Riensche, and P. K. Rao (2022). Discrete Green’s Functions and Spectral Graph Theory for Computationally Efficient Thermal Modeling. International Journal of Heat and Mass Transfer, 183; 122112
Cole, K.D., M. R. Yavari, and P. K. Rao (2020). Computational Heat Transfer with Spectral Graph Theory: Quantitative Verification. International Journal of Thermal Sciences, 153; 106383
Ding, L., C. Li, D. Jin, and S. Ding (2024). Survey of Spectral Clustering Based on Graph Theory. Pattern Recognition, 151; 110366
Dip, S. R. and H. K.Meena (2026). Hardware-Based Brain Tumor Classification Using Graph Laplacian Spectral Features. Brain Research, 1873; 150117
Fitriani, F., I. E.Wijayanti, A. Faisol, and S. Ali (2025). Commuting and Centralizing Maps on Modules. Science and Technology Indonesia, 10(3); 690–697
Fujita, T., A. K. Das, A. Mehmood, S. Das, and V. Duran (2026). Decision Analytics Applications of the Relationship Between TreeSoft Graphs and ForestSoft Graphs. Applied Decision Analytics, 2(1); 73–92
Fujita, T., A. Mehmood, and A. A. Ghaib (2025). Hyperfuzzy OffGraphs: A Unified Graph-Based Theoretical Decision Framework for Hierarchical Under Off-Uncertainty. Applied Decision Analytics, 1(1); 1–14
Gantmacher, F. R. (1959). The Theory of Matrices. Chelsea Publishing Company
Gutman, I. (1978). The Energy of a Graph. Ber. Math. Statist. Sekt. Forschungszentrum Graz, 72; 1–2
Hao, S., G. Zhong, and X. Ma (2022). Notes on the Co-Prime Order Graph of a Group. Proceedings of the Bulgarian Academy of Sciences, 75(3); 340–348
Horn, R. A. and C. R. Johnson (1985). Matrix Analysis. Cambridge University Press
Hu, D. and J. Huo (2026). Concentration of the Adjacency Matrix and of the Normalized Laplacian Matrix in General Random Signed Graphs. Linear Algebra and its Applications, 746; 48–66
Hua, B., A. D. Mednykh, I. A. Mednykh, and L.Wang (2026). On the Complexity of Cayley Graphs on a Dihedral Group. Discrete Mathematics, 349(1); 114662
Krisnawati, V. H., A. N. Hidayat, and M. H. Khuluq (2026). Construction of Binary Linear Codes from Zero Divisor Graphs. Science and Technology Indonesia, 11(3); 784–794
Li, H., G. Zhong, and X. Ma (2022). Finite Groups Whose Co-Prime Order Graphs Have Positive Genus. Proceedings of the Bulgarian Academy of Sciences, 75(9); 1270–1278
Mannan, M. A., H. Akter, and M. A. Ullah (2022). Evaluate All the Order of Every Element in the Higher Even, Odd, and Prime Order of Group for Composition. Science and Technology Indonesia, 7(3); 333–343
Niño, J. and O. K. Johnson (2025). Data-Driven 2D Grain Growth Microstructure Prediction Using Deep Learning and Spectral Graph Theory. Computational Materials Science, 247; 113504
Pirzada, S. and I. Gutman (2008). Energy of a Graph Is Never the Square Root of an Odd Integer. Applicable Analysis and Discrete Mathematics, 2; 118–121
Rada, J. (2026). Extremal Sums of Local Vertex Energies and Graph Energy Decompositions. Linear Algebra and its Applications, 749
Romdhini, M. U., A. Abdurahim, F. Al-Sharqi, S. Bahri, and A. E. S. H. Maharani (2025). On Energy of Prime Coprime Graph for Dihedral Groups. Mathematics and Statistics, 13(3); 175–179
Romdhini, M. U. and A. Nawawi (2026). Relation Between Randič and Harmonic Energies of Commuting Graph for Dihedral Groups. Science and Technology Indonesia, 11(2); 481–488
Saini, M., V. Kumari, P. Rana, A. Sehgal, and D. Singh (2025). Degree Based Matrices of Co-Prime Order Graph of Finite Groups. Mathematics and Statistics, 13(3); 127–135
Saini, M., G. Singh, A. Sehgal, and D. Singh (2024). On Divisor Labeling of Co-Prime Order Graphs of Finite Groups. Italian Journal of Pure and Applied Mathematics, 51; 443–451
Salman, M., T. Noreen, M. U. Rehman, J. Cao, and M. Z. Abbas (2022). Non-Commuting Graph of the Dihedral Group Determined by Hosoya Parameters. Alexandria Engineering Journal, 61(5); 3709–3717
Sehgal, A., Manjeet, and D. Singh (2021). Co-Prime Order Graphs of Finite Abelian Groups and Dihedral Groups. Journal of Mathematics and Computer Science, 23(3); 196–202
Verma, P., S. Nagarajan, and A. Raj (2022). Spectral Graph Theory of Brain Oscillations-Revisited and Improved. NeuroImage, 249; 118919
Adhikari, A. and S. Banerjee (2022). Prime Coprime Graph of a Finite Group. Novi Sad Journal of Mathematics, 52
Al-Tarimshawy, A. S. A. and J. Siemons (2021). Singular Graphs with Dihedral Group Action. Discrete Mathematics, 344(1); 112119
Aschbacher, M. (2000). Finite Group Theory. Cambridge University Press
Bapat, R. B. and S. Pati (2004). Energy of a Graph Is Never an Odd Integer. Bulletin of Kerala Mathematics Association, 1; 129–132
Brouwer, A. E. andW. H. Haemers (2011). Spectra of Graphs. Springer
Codogni, G. and G. M. Lido (2026). Spectral Theory of Isogeny Graphs. Journal of Number Theory, 286; 131–184
Cole, K. D., A. Riensche, and P. K. Rao (2022). Discrete Green’s Functions and Spectral Graph Theory for Computationally Efficient Thermal Modeling. International Journal of Heat and Mass Transfer, 183; 122112
Cole, K.D., M. R. Yavari, and P. K. Rao (2020). Computational Heat Transfer with Spectral Graph Theory: Quantitative Verification. International Journal of Thermal Sciences, 153; 106383
Ding, L., C. Li, D. Jin, and S. Ding (2024). Survey of Spectral Clustering Based on Graph Theory. Pattern Recognition, 151; 110366
Dip, S. R. and H. K.Meena (2026). Hardware-Based Brain Tumor Classification Using Graph Laplacian Spectral Features. Brain Research, 1873; 150117
Fitriani, F., I. E.Wijayanti, A. Faisol, and S. Ali (2025). Commuting and Centralizing Maps on Modules. Science and Technology Indonesia, 10(3); 690–697
Fujita, T., A. K. Das, A. Mehmood, S. Das, and V. Duran (2026). Decision Analytics Applications of the Relationship Between TreeSoft Graphs and ForestSoft Graphs. Applied Decision Analytics, 2(1); 73–92
Fujita, T., A. Mehmood, and A. A. Ghaib (2025). Hyperfuzzy OffGraphs: A Unified Graph-Based Theoretical Decision Framework for Hierarchical Under Off-Uncertainty. Applied Decision Analytics, 1(1); 1–14
Gantmacher, F. R. (1959). The Theory of Matrices. Chelsea Publishing Company
Gutman, I. (1978). The Energy of a Graph. Ber. Math. Statist. Sekt. Forschungszentrum Graz, 72; 1–2
Hao, S., G. Zhong, and X. Ma (2022). Notes on the Co-Prime Order Graph of a Group. Proceedings of the Bulgarian Academy of Sciences, 75(3); 340–348
Horn, R. A. and C. R. Johnson (1985). Matrix Analysis. Cambridge University Press
Hu, D. and J. Huo (2026). Concentration of the Adjacency Matrix and of the Normalized Laplacian Matrix in General Random Signed Graphs. Linear Algebra and its Applications, 746; 48–66
Hua, B., A. D. Mednykh, I. A. Mednykh, and L.Wang (2026). On the Complexity of Cayley Graphs on a Dihedral Group. Discrete Mathematics, 349(1); 114662
Krisnawati, V. H., A. N. Hidayat, and M. H. Khuluq (2026). Construction of Binary Linear Codes from Zero Divisor Graphs. Science and Technology Indonesia, 11(3); 784–794
Li, H., G. Zhong, and X. Ma (2022). Finite Groups Whose Co-Prime Order Graphs Have Positive Genus. Proceedings of the Bulgarian Academy of Sciences, 75(9); 1270–1278
Mannan, M. A., H. Akter, and M. A. Ullah (2022). Evaluate All the Order of Every Element in the Higher Even, Odd, and Prime Order of Group for Composition. Science and Technology Indonesia, 7(3); 333–343
Niño, J. and O. K. Johnson (2025). Data-Driven 2D Grain Growth Microstructure Prediction Using Deep Learning and Spectral Graph Theory. Computational Materials Science, 247; 113504
Pirzada, S. and I. Gutman (2008). Energy of a Graph Is Never the Square Root of an Odd Integer. Applicable Analysis and Discrete Mathematics, 2; 118–121
Rada, J. (2026). Extremal Sums of Local Vertex Energies and Graph Energy Decompositions. Linear Algebra and its Applications, 749
Romdhini, M. U., A. Abdurahim, F. Al-Sharqi, S. Bahri, and A. E. S. H. Maharani (2025). On Energy of Prime Coprime Graph for Dihedral Groups. Mathematics and Statistics, 13(3); 175–179
Romdhini, M. U. and A. Nawawi (2026). Relation Between Randič and Harmonic Energies of Commuting Graph for Dihedral Groups. Science and Technology Indonesia, 11(2); 481–488
Saini, M., V. Kumari, P. Rana, A. Sehgal, and D. Singh (2025). Degree Based Matrices of Co-Prime Order Graph of Finite Groups. Mathematics and Statistics, 13(3); 127–135
Saini, M., G. Singh, A. Sehgal, and D. Singh (2024). On Divisor Labeling of Co-Prime Order Graphs of Finite Groups. Italian Journal of Pure and Applied Mathematics, 51; 443–451
Salman, M., T. Noreen, M. U. Rehman, J. Cao, and M. Z. Abbas (2022). Non-Commuting Graph of the Dihedral Group Determined by Hosoya Parameters. Alexandria Engineering Journal, 61(5); 3709–3717
Sehgal, A., Manjeet, and D. Singh (2021). Co-Prime Order Graphs of Finite Abelian Groups and Dihedral Groups. Journal of Mathematics and Computer Science, 23(3); 196–202
Verma, P., S. Nagarajan, and A. Raj (2022). Spectral Graph Theory of Brain Oscillations-Revisited and Improved. NeuroImage, 249; 118919
Authors
Romdhini, M. U., Abdurahim, Kilicman, A., Pamucar, D., Al-Sharqi, F., Dani, I. H., & Susanto, M. (2026). On the Spectral Characterization of Prime Coprime Graphs Associated with Dihedral Groups. Science and Technology Indonesia, 11(4), 1516–1523. https://doi.org/10.26554/sti.2026.11.4.1516-1523
This work is licensed under a Creative Commons Attribution 4.0 International License.